User x-curious - MathOverflow

When does the sheaf cohomology of a topological space vanish?

2013-04-13T05:28:01Z

The question is in the title. A more precise formulation is:

Let $X$ be a topological space. When does $H^i(X,F) = 0$ for all $i > 0$ and all abelian sheaves $F$ on $X$?

The obvious example is a discrete space. I'd be happy with a characterization of compact Hausdorff topological spaces $X$ satisfying the above property.

Edit: Following Georges Elencwajg's answer, I would like to clarify that these spaces will be quite pathological from the viewpoint of classical topology. Nevertheless, I do not know a single example which does satisfy the above vanishing property and is not discrete. For example, does the Cantor set have this property?

When does the cotangent complex vanish?

2013-01-13T17:40:03Z

The question is already in the title. Less succinctly, let's call a map $f:X \to Y$ of schemes $L$-trivial if its cotangent complex is quasi-isomorphic to $0$. Such maps have striking deformation-theoretic consequences; for example, any deformation of $Y$ can be followed uniquely by a deformation of $X$.

My primary (and probably naive) question is:

Is there a classification of $L$-trivial maps?

I am sure this question has been asked before, but I did not find any literature that deals with it. The three examples of $L$-trivial maps I am familiar with are:

Etale morphisms (and these are the only examples under finiteness constraints).
Any map between perfect $\mathbb{F}_p$-schemes.
The inclusion of the closed point in the spectrum of a valuation ring with divisible value group, or similar "divisible" constructions. For example, $\mathrm{Spec}(\mathbb{C}) \hookrightarrow \mathrm{Spec}(\mathbb{C}[ t^{\mathbb{Q}_{\geq 0}}])$ is $L$-trivial.

[ Edit: I learnt the last one in conversation after positing the first version of this question. ]

More examples can be obtained by taking filtered colimits of the above examples, but those are only slightly different. Hence, a second question is: are there other fundamentally different examples of $L$-trivial maps?

Perhaps a classification is unreasonable to expect, so I am also happy to learn more about $L$-trivial maps in other geometric categories, like algebraic stacks, or derived/spectral schemes/stacks, or (complex/rigid) analytic spaces, etc.. In particular, I am especially curious to know if $L$-trivial maps can be better understood using derived algebraic geometry.

Comment by X-curious

2013-04-13T23:38:00Z

Thanks! If you write this as an answer, I would be happy to accept it.

Comment by X-curious

2013-04-13T17:30:17Z

Thanks for these examples! I should have said that I realize these examples will be "pathological" (and typically very disconnected), and was hoping for some result along the lines that there aren't very many of them besides discrete sets. I will clarify this in the question.

Comment by X-curious

2013-01-14T11:19:39Z

Ah, thanks. This observation strongly supports the (well-advertised) point of view that a simplicial commutative ring is just a ring with additional "nilpotent" data. It also raises the following question: if $A \to B$ is an $L$-trivial map of ordinary rings that is additionally an isomorphism modulo nilpotents, then is $A \simeq B$?

Comment by X-curious

2013-01-13T18:36:57Z

Yes for the second question about finitely presented maps (as I indicated parenthetically in the question), but I am interested in the general case. I do not think $L$-trivial maps are the same as formally etale maps; the latter only corresponds to the vanishing of the first couple of cohomology sheaves of the cotangent complex, and I see no reason why this implies vanishing of the full complex without additional strong finiteness assumptions (like Quillen's conjecture proven by Avramov), but maybe I am missing something simpler?